Liouville theorems for a general class of nonlocal operators

TL;DR

This paper extends Liouville theorems to a broad class of nonlocal linear operators, including anisotropic and nonsymmetric cases, by classifying solutions of the equation  in  with  belonging to this general operator class.

Contribution

It generalizes existing Liouville theorems from fractional Laplacians to a wider class of nonlocal operators, including anisotropic and nonsymmetric types.

Findings

Classified distributional solutions for a broad class of nonlocal operators.

Extended Liouville theorems beyond the fractional Laplacian case.

Provided a unified framework for understanding solutions of nonlocal equations.

Abstract

In this paper, we study the equation $\mathcal{L} u=0$ in $\mathbb{R}^N$, where $\mathcal{L}$ belongs to a general class of nonlocal linear operators which may be anisotropic and nonsymmetric. We classify distributional solutions of this equation, thereby extending and generalizing recent Liouville type theorems in the case where $\mathcal{L}= (-\Delta)^s$, $s \in (0,1)$ is the classical fractional Laplacian.